Alaska's a spendy venture no matter how you go about it.

Most economical trip I ever made there was when I went discovering Alaska by road.
I took a small RV starting in Montana, we were up the haul road and down the Kenai and many points in between.
Slum'ed it all the way cooking most our meals and mostly camping for free.

Spent 60 days, and needed every one of them.
We successfully fished and successfully hunted Moose and Caribou and returned to Montana.
I averaged spending $1250.00 a week for two of us and that was 20 years ago, it was 10K for the 60 days.

You'd barely have time to get started before it was time to leave if you only had a week.
You'd spend more money than you'd ever dreamed possible on rental cars, hotels and restaurants alone.

Some form of and outfitted trip is a good idea when your time is short.